Back to termination

1. Back to termination • So if F is monotonic, we have what we want: finite height ) termination, without the outer join • Also, if the local flow functions are monotonic, then global flow function F is monotonic

2. Another benefit of monotonicity • Suppose Marsians came to earth, and miraculously give you a fixed point of F, call it fp. • Then:

3. Another benefit of monotonicity • Suppose Marsians came to earth, and miraculously give you a fixed point of F, call it fp. • Then:

4. Another benefit of monotonicity • We are computing the least fixed point...

5. Recap • Let’s do a recap of what we’ve seen so far • Started with worklist algorithm for reaching definitions

6. Worklist algorithm for reaching defns let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ; for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0 .. info_out.length do let new_info := m(n.outgoing_edges[i]) [ info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

7. Generalized algorithm using lattices let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ? for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0 .. info_out.length do let new_info := m(n.outgoing_edges[i]) t info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

8. Next step: removed outer join • Wanted to remove the outer join, while still providing termination guarantee • To do this, we re-expressed our algorithm more formally • We first defined a “global” flow function F, and then expressed our algorithm as a fixed point computation

9. Guarantees • If F is monotonic, don’t need outer join • If F is monotonic and height of lattice is finite: iterative algorithm terminates • If F is monotonic, the fixed point we find is the least fixed point.

10. What about if we start at top? • What if we start with >: F(>), F(F(>)), F(F(F(>)))

11. What about if we start at top? • What if we start with >: F(>), F(F(>)), F(F(F(>))) • We get the greatest fixed point • Why do we prefer the least fixed point? • More precise

12. Graphically y 10 10 x

13. Graphically y 10 10 x

14. Graphically y 10 10 x

15. Graphically, another way

16. Another example: constant prop • Set D = in x := N Fx := N(in) = out in x := y op z Fx := y op z(in) = out

17. Another example: constant prop • Set D = 2 { x ! N | x 2 Vars Æ N 2 Z } in x := N Fx := N(in) = in – { x ! * } [ { x ! N } out in x := y op z Fx := y op z(in) = in – { x ! * } [ { x ! N | ( y ! N1 ) 2 in Æ ( z ! N2 ) 2 in Æ N = N1 op N2 } out